Magnetic Neutron Scattering - Appunti

Magnetic Neutron Scattering and Spin

Waves

Tomarchio Luca

February 1, 2019

2

Contents

1 Introduction 5

1.1 Neutron Interaction with Matter . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Magnetic Scattering Theory . . . . . . . . . . . . . . . . . . . . 6

2 Magnetic Scattering by a Crystal 9

2.1 Elastic and Inelastic Magnetic Scattering . . . . . . . . . . . . . . . . . 11

2.1.1 Magnetic Bragg Scattering . . . . . . . . . . . . . . . . . . . . . 13

2.2 Magnons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Anti-Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Triple-Axis Spectrometry (TAS) . . . . . . . . . . . . . . . . . . . . . . 16

3 Appendix 19

3.1 Atomic Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Filled Shells: Larmor Diamagnetism . . . . . . . . . . . . . . . 21

3.1.2 Partially Filled Shells: Paramagnetism . . . . . . . . . . . . . . 21

3.2 Magnetism in Insulating Solids . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 The Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Magnetic Structures . . . . . . . . . . . . . . . . . . . . . . . . 25

3

4 CONTENTS

Chapter 1

Introduction

Much of the understanding of the atomic-scale magnetic structures and the dynamical

properties of solids and liquids are gained from neutron-scattering studies. Because

the neutron has no electric charge, it is an ideal weakly interacting and highly pene-

trating probe of matter’s inner structure. Furthermore, it doesn’t cause deformations

like photon electric fields or charged particles, permitting a determination of the ma-

terial’s intrinsic, unperturbed physical properties. The method is also not sensitive to

extraneous charges, electric fields, and the imperfection of surface layers.

The magnitude of the cross-section of the neutron magnetic scattering is similar to

the cross-section of nuclear scattering by short-range nuclear forces, and is large enough

to provide measurable scattering by the ordered magnetic structures and electron spin

fluctuations [2].

1.1 Neutron Interaction with Matter

The neutron is one of the basic constituents of nuclei together with the proton. Outside

the nuclear contest, a free neutron’s lifetime is only about 15 minutes, after which it

undergoes a β-decay into a proton, an electron, and an antineutrino. Nevertheless, this

lifetime is long enough for neutron-scattering experiments.

Magnetic Gyromagnetic

Electric g-factor g

n

−1

Spin Mass m (g) Moment ratio γ (s /G)

n n −g

charge µ = µ s

n n N n

µ (erg/G) µ = γ σ

n n n n

.

−24 −24 4

× × −1.832 ×

0 1/2 1.675 10 9.662 10 10 3.826

−24

×

Table 1.1: Basic properties of a neutron, µ = e}/(2m c) = 5.05 10 erg/G is the

N p

nuclear magneton and σ (s = σ /}) denotes the nuclear angular momentum.

n n n 5

6 CHAPTER 1. INTRODUCTION

Neutrons used in scattering processes are non-relativistic. Therefore, the neutron’s

energy E is related to its velocity v , wave vector k , and wavelength λ , through

n n n n

2 2 2

2 k h

m v }

n n

n = =

E =

n 2

2 2m 2m λ

n n n

Neutrons used for spectroscopic measurements of matter are slow, meaning that they

'

have a thermal energy E 25 meV . These can be generated through nuclear fission

n

with an associated moderator, or by spallation processes. The monochromatic property

is obtained through the selection from a white beam by the Bragg-reflections in a large

single-crystal monochromator.

Two fundamental interactions govern the scattering of neutrons by an atomic sys-

tem and define the scattering cross-section measured in an experiment. The residual

strong interaction, also known as nuclear force, gives rise to scattering by the atomic

nuclei, while the electromagnetic interaction of the neutron’s magnetic moment with

the sample’s internal magnetic fields gives rise to magnetic scattering. To describe the

former interaction, a characteristic parameter, known as scattering length, is intro-

duced. It takes the form −

b (r , R) = b δ(r R)

N n N n

where r is the coordinate of a neutron and R is that of a nucleus. The locality con-

n

dition is due to the short characteristic distances of the nuclear interaction associated.

In the Born approximation, this process can be described through an effective neutron-

nucleus interaction potential 2

2π} −

− b δ(r R) (1.1)

V (r , R) = N n

N n m n

generally known as Fermi pseudopotential. In general, the bound scattering length b N

is considered to be a complex parameter, and, being a phenomenological parameter, it

is determined experimentally.

1.1.1 Magnetic Scattering Theory

The main contribution to magnetic scattering arises from neutron’s interaction with

the total dipole magnetic moment of the atomic electrons; all other electromagnetic

interactions are at least two orders of magnitude smaller and can be neglected [2]. The

fundamental starting point for the evaluation of the magnetic scattering length is the

Pauli Hamiltonian of the electrons under the magnetic field generated by the neutron

spin. The probing particle can be treated as a point dipole with magnetic moment

1.1. NEUTRON INTERACTION WITH MATTER 7

µ = γ σ , with σ = as its angular momentum. Thus, the corresponding expres-

}s

n n n n n

sion for the vector potential at the electron position r takes the form

i

×

µ r

µ n

n

− ∇ × |r − |

=

A (r r ) = , r = r (1.2)

n i n i n

3

r r

where r and r are respectively the electron and neutron positions. Inserting it into

i n

the equation (3.3), it is possible to find the interacting contributions as

e e

1 1

2 2

(i)

H − ·

p + (A + A ) p + A

(r , r ) = + 2µ s B

i n e i e

i n B i

int 2m c 2m c

e

1

· ×

· + s (∇ A ) (1.3)

A p + A

= 2µ i n

n i e

B c

} m

neglecting the diamagnetic term of second order in µ = µ . A denotes the

e

N B e

m p

additional contribution to the total vector potential from the surrounding electrons, or

an external magnetic field.

Given the non-relativistic behaviour of neutrons, we may write the state vector for

the initial plane-wave propagation of neutrons as

1 ik·r

|ks i |s i

= e (1.4)

n

n n

V

When passing through the target, the probability per unit time that a neutron makes a

0 0

|k i

transition from its initial state to the state s is determined by the Fermi’s Golden

n

Rule: 2π X 0 0

0 0 2

|k i| −

P (β)|hks ; i|H s ; f δ(}ω + E E ) (1.5)

W (ks , k s ) = i n int n i f

n n } f,i

where is the energy transferred, to which is associated a momentum exchange

}ω 0

−

= , where q is known as scattering vector. The summation is developed

}q }k }k

over the final and initial target states, the latter weighted through a Boltzmann factor.

0

The scattered neutrons with momenta lying in a narrow range around are counted

}k

0

by placing a detector in a direction along k , subtending a small element of solid angle

0

dΩ. The value of k and the final neutron energy are determined, again, by making

use of Bragg-reflection in a single crystal analyser, so that only neutrons with energy

0 2

energies in a small interval dE around (}k ) /2m strike the counter. Their number

n

per unit time and per incident neutron is proportional to the differential scattering

8 CHAPTER 1. INTRODUCTION

cross-section 0

2

∂ σ k m

X

n (i) 0 2

i| −

= P (β)|hs ; i|H (q)|s ; f δ(}ω + E E ) (1.6)

i n n i f

int

2

∂E∂Ω k 2π} i,f

where Z

(i) (i) −iq·r

H H −

(q) = (r r )e dr

n

i n n

int int 1

This result of time-dependent perturbation theory, in the first Born approximation ,

is accurate because of the very weak interaction between the neutrons and the con-

stituents of the sample. The Fourier transform of A with respect to the neutron

n

coordinate is

Z Z 4π

−iq·r −iq·r −iq·r iq·r ×

− −e µ q̂

A (r r )e dr = e A (r)e dr =

n i i n

n i n n n iq

and we can also compute

Z Z

−iq·r −iq·r

× − × × ×

(∇ A (r))e dr = (∇e ) A (r)dr = 4πq̂ µ q̂

n n n

−iq·r

∇ × → ∞.

where the second integral with (e A (r)) goes to zero in the limit r

n

From these results we obtain

i

(i) −iq·r 0

H × · · × ×

(q) = 2µ e 4π µ q̂ p + s (q̂ µ q̂) (1.7)

i

B n i n

i

int }q

i 0 −iq·r

×

· × ×

q̂ p

= 8πµ µ + q̂ s q̂ e i

B n i

i

}q

0 ec A .

where p = p + e

1 Given the scattered wave function at great distances from the scatterer centre r to be

0

ikr

e

ikz

ψ(r) = Ae + A f (k)

r

with Z

m

n −ir·r

− H

f (k) = e (r )ψ(r )dr

0 int 0 0 0

2

2πA}

the scattering amplitude, the Born approximation consists into taking the condition that the potential

ikz

does not significantly alter the wave function: ψ(r ) = ψ (r ) = Ae .

0 0 0

Chapter 2

Magnetic Scattering by a Crystal

The Hamiltonian of the neutron’s magnetic interaction with a crystal is the sum of the

interactions (1.3), where r is replaced by R + r , over the lattice position R where

i j ij j

the magnetic atoms are located. Equation (1.7) may then be written as

(i,j) −iq·R

·

H (q) = 8πµ µ (Q + Q )e j

B n p s

int

introducing i 0 −iq·r −iq·r

× × ×

Q = q̂ p e ; Q = q̂ s q̂e

ij ij

p s i

i

}q −iq·r

hi|Q |f i,

In order to calculate the matrix elements the factor e is expanded in

p,s

· ·

spherical Bessel functions j (ρ), with ρ = q r and cos θ = q r/ρ

n

∞

X

−iq·r n ' − ·

e = (2n + 1)(−i) j (ρ)P (cos θ) j (ρ) iq r[j (ρ) + j (ρ)] (2.1)

n n 0 0 2

n=0

with the truncation valid for small values of ρ. Introducing this expansion in the ex-

pression for Q , we find

p 1 0 0 0 0

× × ×

[j (ρ) + j (ρ)]q̂ l q̂ + Q = r p

Q = }l

p 0 2 p

2

where we have introduced

i 1

0 0 0 0

× · ·

Q = q̂ j (ρ)p + [j (ρ) + j (ρ)][(q̂ r)p + (q̂ p )r] + ...

0 0 2

p 2}

}q

H

I